I'm a little surprised that no one answered with the following:

An incompact set of first order sentences.

That would be, a set of sentences whose finite subsets have models, but there is no model of the whole set. As a particular example, one could say that an incompact sentence is one having models of arbitrarily high finite cardinalities but no infinite models.

Vaguely related to the previous example, in my specific field I'm interested in proving that certain formulas (do not) characterize directly indecomposable structures. Most of the time, you might take two very similar finite structures $\mathbf{A}$ and $\mathbf{B}$, such that $\mathbf{A}$ has direct product decomposition, and $\mathbf{B}$ results of a minor tweak of $\mathbf{A}$ (such as adding or dropping some elements), in such a way $|\mathbf{B}|$ is prime, hence indecomposable. The relevant entity here is

A nontrivial factorization of a prime number.